Nonlinear Dynamical Systems And Orthogonal Polynomials

To be able to describe the dynamical behaviour of discrete dynamical systems for rational functions in the complex plane a good knowledge of the Julia set, that is the set where the function is chaotic, is necessary. One of the main investigations of the project was how to conclude from properties of the iterated function to the structure of the Julia set as for instance to the connectedness. We suceeded in giving constructive necessary and sufficient conditons which guarantee the connectedness of the Julia set for rational functions. Furthermore we were able to give an explicite representation of that rational functions which have a Jordan arc as Julia set. Of particular interest are functions which are chaotic on the whole Riemann sphere. As is known functions whose critical points are preperiodic have this property, but by theoretical investigations it is known also that the class of functions is much wider. We found now constructive simple conditions sufficient for the Julia set to be the whole Riemann sphere. Further we have shown how to get by composition wide classes of functions with this property. Many dynamical systems as Toda lattices, nonlinear Schrödinger equations, KdV equations etc. can be solved explicitely or approximately by orthonormal polynomials. Nowadays of foremost interest are such dynamical systems whose spectrum consists of nonconnected sets, as several intervals, Cantor sets, Julia sets etc. . For these reasons one would like to have asymptotic representations of orthonormal polynomials on such sets. By a new approach using Hardy spaces of character automorphic functions we obtained asymptotic representations of polynomials orthonormal on homogenous sets (which include several intervals, thick Cantor sets etc.). Such asymptotic representations have been found on and off the spectrum from which asymptotics for the recurrence coefficients of the orthonormal polynomials follow also. By different methods (methods of inverse polynomial mappings) we obtained some results for polynomials orthonormal on Julia sets also. In the case of Toda-lattices the connection with orthonormal polynomials is particularily simple. Indeed, the particles of the Toda lattice can be considered as the recurrence coefficients of orthonormal polynomials as follows by Flaschka`s transformation. Thus the question arises which orthogonality measures give us solutions of Toda lattices. For stationary initial conditions such a characterization has been given by Kac - VanMoerbeke. Under general inital conditions this question is still open. By considering instead of the measure the associated Stieltjes function we demonstrated that the recurrence coefficients are the solution of a Toda lattice if and only if the Stieltjes function satisfies a Ricatti-type differential equation.